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International Journal of Rock Mechanics & Mining Sciences 45 (2008) 1226–1236

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How sorption-induced matrix deformation affectsgas flow in coal seams: A new FE model

Hongbin Zhanga,c, Jishan Liua,�, D. Elsworthb

aSchool of Oil and Gas Engineering, The University of Western Australia, WA 6009, AustraliabDepartment of Energy and Geo-Environmental Engineering, Penn State University, USA

cHainan University, Hainan Province, PR China

Received 21 December 2006; received in revised form 28 November 2007; accepted 30 November 2007

Available online 15 January 2008

Abstract

The influence of sorption-induced coal matrix deformation on the evolution of porosity and permeability of fractured coal seams is

evaluated, together with its influence on gas recovery rates. The porosity-based model considers factors such as the volume occupied by

the free-phase gas, the volume occupied by the adsorbed phase gas, the deformation-induced pore volume change, and the sorption-

induced coal pore volume change. More importantly, these factors are quantified under in situ stress conditions. A cubic relation between

coal porosity and permeability is introduced to relate the coal storage capability (changing porosity) to the coal transport property

(changing permeability). A general porosity and permeability model is then implemented into a coupled gas flow and coal deformation

finite element model. The new FE model was used to compare the performance of the new model with that of the Palmer–Mansoori

model. It is found that the Palmer–Mansoori model may produce significant errors if loading conditions deviate from the assumptions of

the uniaxial strain condition and infinite bulk modulus of the grains. The FE model was also applied to quantify the net change in

permeability, the gas flow, and the resultant deformation in a coal seam. Model results demonstrate that the evolution of porosity and of

permeability is controlled by the competing influences of effective stresses and sorption-based volume changes. The resulting sense of

permeability change is controlled by the dominant mechanism.

r 2008 Elsevier Ltd. All rights reserved.

Keywords: Matrix shrinking; Coal permeability; Coal porosity; Coupled model; Simulation

1. Introduction

Methane in coal seams is an important natural energyresource, although ignition and the resulting explosionhazard remain a major problem during coal mining.Degassing seams is an important method to mitigate thishazard, and results in the beneficial recovery of a clean-burning and low-carbon fuel resource. The injection ofcarbon dioxide to preferentially dissociate methane hasbeen an effective measure used after primary recovery bydepressurization. Recently, carbon dioxide (CO2) seques-tration in deep coal seams has attracted more attention as amethod of reducing the output of greenhouse gases to theatmosphere [1].

e front matter r 2008 Elsevier Ltd. All rights reserved.

mms.2007.11.007

ing author. Tel.: +618 6488 7205; fax: +61 8 6488 1964.

ess: Jishan@cyllene.uwa.edu.au (J. Liu).

Gas flow within coal seams is quite different from that ofconventional reservoirs. Detailed studies have examinedthe storage and transport mechanisms of gas in coal seams.In situ and laboratory data indicate that the storage andflow of gas in coal seams is associated with the matrixstructure of coal and the absorption or desorption of gas.Coal is a naturally fractured dual-porosity reservoir [2],consisting of micro-porous matrix and cleats. Most of thegas is initially stored within micro-pores in the absorbedstate [3]. When gas recovery begins, the gas desorbs anddiffuses from matrix to cleats due to the concentrationgradient. The gas flowing through the cleats is consideredto be gas seepage controlled by the permeability of the coalseam [4]. Experimental results have shown that gassorption generally follows a Langmuir isotherm [5,6].Desorption plays an important role in both defining thelongevity and rate of the gas supply, and in controlling the

ARTICLE IN PRESSH. Zhang et al. / International Journal of Rock Mechanics & Mining Sciences 45 (2008) 1226–1236 1227

related deformation of the solid matter comprisingthe seam.

A variety of experiments have investigated sorptioncharacteristics under isothermal conditions [4,7–10] withsupporting models representing isothermal response[8,9,11–13]. These studies have also noted the dependencyof volumetric strain of the coal matrix as a non-linearfunction of gas pressure, driven by gas desorption. There isan approximately linear relationship between the sorption-induced volumetric strain and the absorbed gas volume[7,9,10]. This relation holds both during uptake, describedas sorption, and in discharge, described as desorption, ofgases from the surfaces of the coal matrix. Because of thedual-porosity structure of coal seams (i.e., micro-porousmatrix and macro-porous cleat/fracture network), the coalmatrix represents the main reservoir for the gas, and thecleats the main fracture pathways. When the pore pressuredeclines during methane production, methane desorbsfrom the coal matrix and the desorbed gas flows throughthe cleats to the producing well. The decline of porepressure results in a concomitant increase in effective stress.The increase in effective stress reduces the stress-sensitivepermeability of the cleat system. In contrast, the deso-rption-induced shrinkage of the coal matrix widens thecleats and enhances permeability. The net change inpermeability accompanying gas production is thus con-trolled by the competitive effects of declining pore pressuredecreasing permeability, and the shrinking coal matrixincreasing permeability. The net effect, of permeability lossor permeability gain, is dependent on the mechanicalboundary conditions applied to the system.

Adsorption of gas, such as carbon dioxide, is the reverse ofdesorption. When the gas pressure increases, the gas adsorbsonto the coal matrix. The increase of pore pressure results ina decrease of effective stress. The reduction in effective stressenhances the coal permeability. In contrast, the adsorption-induced swelling of the coal matrix reduces the cleatapertures and decreases the permeability. The net changein permeability accompanying gas sequestration is alsocontrolled competitively by the influence of effective stressesand matrix swelling, again controlled by the boundaryconditions applied locally between the end-members of nullchanges in either mean stress or volume strain.

Numerical simulations of gas diffusion, gas flow, andcoupled hydromechanical response have been widelyapplied. Finite element methods and a formulation formodeling mass transport problems in porous media havebeen applied, including the effects of coupled solid–gasresponse for gas flow in coal seams [14]. This included onlythe effect of gas sorption on mass storage [15]. Valliappanand Zhang developed a coupled model incorporating theeffect of diffusion of adsorbed methane gas from the solidmatrix to the voids [16]. A dual-porosity poroelastic modelwas extended and utilized in solving generalized planestrain problems [17,18]. Gilman and Beckie proposed asimplified model of methane diffusion and transport in acoal seam and found a reference time of methane release

from the coal matrix into cleats to have a critical influenceon overall methane production [19]. A model for multi-phase flow, coupled with heat transfer and rock deforma-tion, was used to simulate CO2 injection into a brineformation by Rutqvist and Tsang [20]. In 2004, a non-linear coupled mathematical model of solid deformationand gas seepage was presented and the methane extractionfrom fractured coal seam was simulated [21]. However, theconstitutive relationships between stress and strain aresimilar to conventional poroelastic mechanics in most ofthe above simulations and the effect of sorption-inducedstrain on matrix volumetric strain has not been taken intoaccount although experimental data have noted itssignificant impact—both on total volumetric strain of theseam, and the resulting feedback on permeability.The gas flow in coal seams is a complex physical and

chemical process coupling solid deformation, gas deso-rption and gas movement. Although the influence ofsorption-induced deformation on porosity, and on perme-ability has been widely studied, how this in turn affects gasflow within the seam is not well understood. This is partlybecause no coupled gas flow and sorption-induced coaldeformation models are available for in situ stressconditions. The primary motivation of this study is toinvestigate how sorption-induced coal matrix deformationaffects the gas flow in a coal seam through developing sucha porosity-based model.

2. Governing equations

In the following, a set of field equations are definedwhich govern the deformation of the solid matrix, andprescribe the transport and interaction of gas flow in asimilar way to poroelastic theory [22]. These derivationsare based on the following assumptions: (a) coal is ahomogeneous, isotropic and elastic continuum. (b) Strainsare much smaller than the length scale. (c) Gas containedwithin the pores is ideal, and its viscosity is constant underisothermal conditions. (d) The rate of gas flow through thecoal is defined by Darcy’s law. (e) Conditions areisothermal. (f) Coal is saturated by gas. (g) Compositionsof the gas are not competitive, i.e., one gas component ata time.

2.1. Governing equation for coal seam deformation

For a homogeneous, isotropic and elastic medium, thestrain-displacement relation is expressed as

�ij ¼12ðui;j þ uj;iÞ, (1)

where eij is the component of the total strain tensor and ui isthe component of the displacement. The equilibriumequation is defined as

sij;j þ f i ¼ 0, (2)

where sij denotes the component of the total stress tensorand fi denotes the component of the body force. The

ARTICLE IN PRESSH. Zhang et al. / International Journal of Rock Mechanics & Mining Sciences 45 (2008) 1226–12361228

following conventions have been adopted in Eqs. (1) and(2) and following related equations: a comma followed bysubscripts denotes differentiation with respect to spatialcoordinates and repeated indices in the same monomialimply summation over the range of the indices (generally1–3, unless otherwise indicated) [23].

The gas sorption-induced strain es is presumed to resultin volumetric strain only. Its effects on all three normalcomponents of strain are same. On the basis of poroelas-ticity [23] and by making an analogy between thermalcontraction and matrix shrinkage [12], the constitutiverelation for the deformed coal seam becomes

�ij ¼1

2Gsij �

1

6G�

1

9K

� �skkdij þ

a3K

pdij þ�s3dij, (3)

where G ¼ E/2(1+n), K ¼ E/3(1�2n), a ¼ 1�K/Ks, andskk ¼ s11+s22+s33, where K is the bulk modulus of coal,Ks is the bulk modulus of the coal grains, G is the shearmodulus of coal, E is the Young’s modulus of coal, n isPossion’s ratio of coal, a is the Biot coefficient, dij is theKronecker delta, and p is the gas pressure within the pores.From Eq. (3), we obtain

�v ¼ �1

Kðs̄� apÞ þ �s, (4)

where ev ¼ e11+e22+e33 is the volumetric strain of coalmatrix and s̄ ¼ �skk=3 is the mean compressive stress. Thecomponent of effective stress seij is also defined asseij ¼ sij+apdij. Combination of Eqs. (1)–(3) yields theNavier-type equation expressed as

Gui;kk þG

1� 2nuk;ki � ap;i � K�s;i þ f i ¼ 0. (5)

Eq. (5) is the governing equation for coal deformation,where the gas pressure, p, can be solved from the gas flowequation as discussed following.

2.2. Governing equation for gas flow

The equation for mass balance of the gas is defined as

qm

qtþ r � ðrgqgÞ ¼ Qs, (6)

where rg is the gas density, qg is the Darcy velocity vector,Qs is the gas source or sink, t is the time, and m is the gascontent including free-phase gas and absorbed gas [1], isdefined as

m ¼ rgfþ rgarcVLp

pþ PL, (7)

where rga is the gas density at standard conditions, rc is thecoal density, f is porosity, VL represents the Langmuirvolume constant, and PL represents the Langmuir pressureconstant. According to the ideal gas law, the gas density isdescribed as

rg ¼Mg

RTp, (8)

where Mg is the molecular mass of the gas, R isthe universal gas constant, and T is the absolute gastemperature.Assuming the effect of gravity is relatively small and can

be neglected, the Darcy velocity, qg, is given by

qg ¼ �k

mrp, (9)

where k is the three permeability of the coal and m is thedynamic viscosity of the gas. Substituting Eqs. (7)–(9) intoEq. (6), we obtain

fþrcpaVLPL

ðpþ PLÞ2

� �qp

qtþ p

qfqt� r �

k

mprp

� �¼ Qs, (10)

where pa is one atmosphere of pressure (101.325 kPa). InEq. (10), the permeability k is dependent on the porosity,f, while f is a function of the volumetric strain, en, and thesorption-induced strain, es. Therefore, Eqs. (5) and (10) arecoupled through the porosity–permeability relation.

2.3. A general porosity model

The sorption-induced volumetric strain es is fitted ontoLangmuir-type curves and has been verified throughexperiments [7,9,10]. A Langmuir-type equation is usedto calculate this volumetric strain, defined as

�s ¼ �Lp

PL þ p, (11)

where the Langmuir volumetric strain, eL, is a constantrepresenting the volumetric strain at infinite pore pressureand the Langmuir pressure constant, PL, representing thepore pressure at which the measured volumetric strain isequal to 0.5eL. The authors in the above experiments alsoassume that the coal permeability varies with porosity asfollows:

k

k0¼

ff0

� �3

, (12)

where the subscript, 0, denotes the initial value of thevariable. The porosity is calculated as a function of coalmechanical properties such as modulus, sorption isothermparameters and pore pressure. However, different studieshave presented different formulae to calculate the coalporosity and permeability.

2.3.1. Review of porosity and permeability models

In the following, we briefly explain each of the modelscurrently available in the literature, and refer the reader tothe source papers for further details.

2.3.1.1. Seidle–Huitt model. This model does not includethe elastic strain of the coal and assumes that allpermeability changes are caused by the sorption-inducedstrain only. Under these assumptions, the porosity and

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permeability are defined as [2]

ff0

¼ 1þ�L3

1þ2

f0

� �p0

PL þ p0

�p

PL þ p

� �, (13)

k

k0¼ 1þ

�L3

1þ2

f0

� �p0

pL þ p0

�p

pL þ p

� �� �3. (14)

2.3.1.2. Palmer–Mansoori model. Unlike the Seidle–Huitt model, the Palmer–Mansoori model considers theelastic deformation of coal under uniaxial stress conditions.The porosity model is defined as [11]

ff0

¼ 1þ1

Mf0

ðp� p0Þ þ�L3f0

K

M� 1

� �p

PL þ p�

p0

PL þ p0

� �,

(15)

k

k0¼ 1�

1

Mf0

ðp� p0Þ þ�Lf0

K

M� 1

� �p

PL þ p�

p0

PL þ p0

� �� �3,

(16)

where M ¼ E(1�n)/(1+n)(1�2n).

2.3.1.3. Shi–Durucan model. The assumptions are sameas the Palmer–Mansoori model [12]:

k

k0¼ exp 3cf

n1� n

ðp� p0Þ þ�L3

E

1� n

� ���

�p0

PL þ p0

�p

PL þ p

� ���, ð17Þ

where cf is cleat volume compressibility.

2.3.1.4. Cui–Bustin model. This model has a general formbut it is only applied to the same assumed situation as thePalmer–Mansoori model [9]:

ff0

¼ exp1

K�

1

Kp

� �½ðs� s0Þ � ðp� p0Þ�

� �, (18)

k

k0¼ exp �

3

Kp½ðs� s0Þ � ðp� p0Þ�

� �, (19)

where Kp is the bulk modulus of pores.

2.3.1.5. Robertson–Christiansen model. In this model, thedeformation of coal grains is neglected and equal axial andradial stresses are assumed [13].

k

k0¼ exp �3c0

1� exp½acðp� p0Þ�

aþ

9

f0

1� 2nEðp� p0Þ

��

��L3

PL

PL þ p0

� �In

PL þ p

PL þ p0

� ���, ð20Þ

where c0 is the initial fracture compressibility and ac is thechange rate in fracture compressibility.

In all these models, the total stress, s, has been assumedas constant, i.e.,

ds ¼ dse � dp ¼ 0 ða ¼ 1Þ, (21)

where se denotes effective stress. Or under invarianttotal stress.

dse ¼ dp. (22)

According to the principle of effective stress, the inducedcoal deformation is determined by the change in effectivestress, dse, which can be replaced by the change in porepressure, dp, under the assumption of null change in totalstress. This is why terms representing effective stress ortotal stress are absent in all of these existing permeabilitymodels. However, this condition can be violated in anumber of circumstances, including the penetration by drillholes, the massive stimulation by the injection of fluids,and the proximity of excavation surfaces. These factorsresult in the re-distribution of total stresses in coal seams.Therefore, a new porosity and permeability model undervariable stress conditions is needed to quantify the gas flowin coal seams.

2.3.2. The general porosity model

Considering a porous medium containing solid volumeof Vs and pore volume of Vp, we assume the bulkvolume V ¼ Vp+Vs and the porosity f ¼ Vp/V. Accordingto Eq. (4), the volumetric evolution of the porous mediumwith the load of s̄ and p can be described in terms of DV/Vand DVp/Vp, the volumetric strain of coal and volumetricstrain of pore space, respectively [23]. The relations are

DV

V¼ �

1

Kðs̄� apÞ þ �s, (23)

DVp

Vp¼ �

1

Kpðs̄� bpÞ þ �s, (24)

where b ¼ 1�Kp/Ks.We assume that the sorption-induced strain for the coal

is the same as for the pore space. Without the gas sor-ption effect, the volumetric variation of the porous med-ium satisfies the Betti–Maxwell reciprocal theorem [23],ðqV=qpÞs̄ ¼ ðqVp=qs̄Þp, and we obtain

Kp ¼fa

K . (25)

Using the definition of porosity, the following expressionscan be deduced as

DV

V¼

DV s

V sþ

Df1� f

, (26)

DVp

Vp¼

DV s

V sþ

Dffð1� fÞ

. (27)

Solving Eqs. (23), (24), (26) and (27), we obtain

Df ¼ f1

K�

1

Kp

� �ðs̄� pÞ. (28)

Substituting Eqs. (4) and (25) into Eq. (28) yields

Df ¼ ða� fÞ �v þp

K s� �s

� �. (29)

ARTICLE IN PRESSH. Zhang et al. / International Journal of Rock Mechanics & Mining Sciences 45 (2008) 1226–12361230

If the initial porosity is f0 at pressure p0 and the initialvolumetric strain is zero, the porosity can be expressed as

f ¼1

1þ S½ð1þ S0Þf0 þ aðS � S0Þ�, (30)

where S ¼ en+(p/Ks)�es, S0 ¼ (p0/Ks)�eLp0/(p0+pL).Considering the cubic law relation, Eq. (12), between

permeability and porosity of the porous media, we obtain

k

k0¼

1

1þ Sð1þ S0Þ þ

af0

ðS � S0Þ

� �� �3

, (31)

where k0 is the initial permeability at the initial pressure p0and porosity f0.

Eqs. (30) and (31) present a general porosity model and ageneral permeability model, respectively. These models canbe applied to variable stress conditions. If we consider S51and S051, the simplified expression for porosity is derived as

f ¼ f0 1þaf0

�v þp� p0

K sþ

�LPLðp0 � pÞ

ðp0 þ PLÞðpþ PLÞ

� �� �. (32)

It is clear that the porosity and permeability of the coal iscontrolled by the matrix volumetric strain associated witheffective stress (Eq. (4)), the grain volumetric strain and thegas desorption-induced volumetric strain. It should be notedthat neither matrix volumetric strain nor effective stress isindependent of gas desorption-induced strain according toEq. (4). It is apparent that the general porosity andpermeability model is coupled with the coal seam deforma-tion. Both the porosity model and the permeability modeldefine the interactions between coal deformation and gasflow.

If S51, S051, and K sbK , the coal seam is underconditions of uniaxial strain, and the overburden load isunchanged, a simplified expression of porosity can bederived from Eq. (30) as

f ¼ f0 þð1þ nÞð1� 2nÞ

Eð1� nÞðp� p0Þ

�2ð1� 2nÞ3ð1� nÞ

�Lp

pþ PL�

�Lp0

p0 þ PL

� �ð33Þ

which is the same as the model presented by Palmer andMansoori [11]. Using the stress–strain relation and assum-ing e33 is the direction of uniaxial strain and overburdenload, the Palmer–Mansoori model can also be expressed as

f ¼ f0 1þ1

f0

�33 þ�LPLðp0 � pÞ

ðp0 þ PLÞðpþ PLÞ

� �� �. (34)

Comparing Eq. (34) with Eq. (32), the Palmer–Mansoorimodel is only applicable to conditions of uniaxial strain,constant overburden load, and infinite bulk modulus of thegrains.

2.4. Coupled governing equations

Substituting Eq. (11) into Eq. (5), we rewrite thegoverning equation for coal seam deformation as

Gui;kk þG

1� 2nuk;ki � ap;i �

K�LPL

ðpþ PLÞ2

p;i þ f i ¼ 0. (35)

From Eq. (30), the partial derivative of f with respect totime is expressed as

qfqt¼

a� f1þ S

q�v

qtþ

1

Ks

qp

qt�

�LPL

ðpþ PLÞ2

qp

qt

� �. (36)

Substituting Eq. (36) into Eq. (10) yields the governingequation for gas flow through a coal seam with gassorption as

fþrcpaVLPL

ðpþ PLÞ2þða� fÞpð1þ SÞK s

�ða� fÞ�LPLp

ð1þ SÞðpþ PLÞ2

� �qp

qt

� r �k

mprp

� �¼ Qs �

ða� fÞpð1þ SÞ

q�v

qt. ð37Þ

The first term on the left-hand side of Eq. (37) representsall the controlling factors on porosity, including thevolume occupied by the free-phase gas, the volumeoccupied by the adsorbed phase gas, the coal mechanicaldeformation-induced pore volume change, and the sorp-tion-induced coal pore volume change. More importantly,these factors are quantified under in situ stress conditions.The second term on the left-hand side is associated with thecharacteristics of gas migration. On the right-hand side, thesecond term is a coupled term including the rate change inthe volumetric strain due to coal deformation. Its contri-bution to the equation can be treated as a source or sinkfrom the mechanical deformation. Therefore, Eqs. (35)–(37) define the coupled gas flow and coal seam deformationmodel.

2.5. Boundary and initial conditions

For the Navier-type Eq. (35), the displacement and stressconditions on the boundary are given as

ui ¼ ~uiðtÞ on qO, (38)

sijnj ¼ ~FiðtÞ on qO, (39)

where ~ui and ~Fi are the known displacements and stresseson the boundary qO, and nj is the unit vector normal to theboundary. The initial conditions for displacement andstress in the domain are described as

uið0Þ ¼ u0 in O, (40)

sijð0Þ ¼ s0 in O, (41)

where u0 and s0 are initial values of displacement and stressin the domain O.For the gas flow Eq. (37), the Dirichlet and Neumann

boundary conditions are defined as

p ¼ ~pðtÞ on qO, (42)

nk

mrp ¼ ~QsðtÞ on qO, (43)

ARTICLE IN PRESS

0.1

m

0.1 m

p(0) = 6.2 MPa (Ω)

Fig. 1. Example I: simulation model of the gas desorption from a coal

sample under the uniaxial plane strain state.

Table 1

Property parameters of Examples I, II, and III

Parameter Value

Young’s modulus of coal, E (MPa) 2713

Young’s modulus of coal grains, Es (MPa) 8139

Possion’s ratio of coal, n 0.339

Density of coal, rc (kg/m3) 1.25� 103

Density of methane, rg (kg/m3) at standard condition 0.717

Methane dynamic viscosity, m (Pa s) 1.84� 10�5

Langmuir pressure constant, PL (MPa) 6.109

Langmuir volume constant, VL (m3/kg) 0.015

Langmuir volumetric strain constant, eL 0.02295

Initial porosity of coal, f0 0.00804

Initial permeability of coal, k0 (m2) 3.7996� 10�17

H. Zhang et al. / International Journal of Rock Mechanics & Mining Sciences 45 (2008) 1226–1236 1231

where ~pðtÞ and ~QsðtÞ are the specified gas pressure and gasflux on the boundary, respectively. The initial condition forgas flow is

pð0Þ ¼ p0 in O. (44)

3. Finite element implementations

The above governing equations, especially the gas flowequation incorporating the effect of desorption, are a set ofnon-linear partial differential equations (PDE) of secondorder in space and first order in time. The non-linearityappears both in the space and time domains; and therefore,these equations are difficult to solve analytically. There-fore, the complete set of coupled equations is implementedinto, and solved by using COMSOL Multiphysics, apowerful PDE-based multiphysics modeling environment.

4. Simulation examples

In the following, we present three simulation examples toillustrate the resultant effects of coupled gas sorption andcoal deformation. These three examples are under differentboundary conditions which causes different stress states.The first one is under uniaxial stress condition. The secondone is under constrained plane strain condition. The lastone is under unconstrained plane strain condition. We usethe three examples to quantify the net change in perme-ability, in gas flow, and in coal deformation accompanyinggas production. These processes are controlled competi-tively by increases in effective stresses and matrixshrinkage. The results are also compared with Palmer–Mansoori model, respectively, to show the limitation ofassumptions in Palmer–Mansoori model.

4.1. Example I: gas desorption under uniaxial stress

conditions

In this example, we follow methane desorption from a coalsample under conditions of uniaxial strain. This geometryrepresents some experimental conditions prescribed in pre-vious published studies and is used to describe the essentialcharacteristics of the gas desorption from coal. The sensitivityof the controlling parameters, including matrix volumetricstrain, porosity, permeability and pore pressure, to the gasdesorption and the ration of the bulk modulus of coal matrixto that of coal grains (K/Ks) were investigated in detail.

The model geometry of 0.1m� 0.1m is shown in Fig. 1.The right side is free to displace while other three sides areconstrained. The pressure on the right side is specified as101.325 kPa. Zero fluxes on the other three sides arespecified. The initial gas pore pressure in the coal is set at6.2MPa. The coal properties are listed in Table 1. Most ofthe parameters were chosen from the experimental results[10], and unreported parameters were substituted fromcontemporary literature.

We present the model results in terms of (1) thecontributions of different volumetric strains to the totalvolumetric strain; (2) the contributions of different gasstorage terms to the total gas storage capability; (3) theevolution of permeability ratios; (4) the effect of coal bulkmodulus ratios on the permeability and the comparisonwith the PM model; (5) the effect of gas desorption on thegas pressure distribution; (6) the effect of coal bulkmodulus ratios on the gas pressure distribution; and (7)the evolution of coal deformation. These results are shownin Figs. 2–8.

4.1.1. Porosity and volumetric strains

As shown in Fig. 2, the porosity varies with thevolumetric strains of coal matrix, grains, and gas sor-ption. When pore pressure declines, the sorption-induced

ARTICLE IN PRESS

-80.0

-60.0

-40.0

-20.0

0.0

20.0

40.0

60.0

80.0

100.0

0.0 1.0 2.0 3.0 4.0 5.0 6.0

Pore Pressure (MPa)

Contr

ibution t

o P

oro

sity R

atio (

%)

Matrix Volumetric

Strain

Grains Volumetric

StrainSorption-Induced

Volumetric Strain

Fig. 2. Example I: contributions of different volumetric strain compo-

nents to the coal porosity ratio.

-10

0

10

20

30

40

50

60

70

80

90

100

0.0 1.0 2.0 3.0 4.0 5.0 6.0

Pore Pressure (MPa)

Contr

ibution t

o S

tora

ge C

oeffic

ient

(%)

term 1

term 2

term 3

term 4

Fig. 3. Example I: contributions of four terms in gas storage coefficient,

i.e., the volume occupied by the free-phase gas (term 1), the volume

occupied by the adsorbed gas phase (term 2), the mechanical deformation-

induced pore volume change (term 3), and the sorption deformation-

induced pore volume change (term 4).

0.9

1.0

1.1

1.2

1.3

1.4

1.5

1.6

0.00 0.02 0.04 0.06 0.08 0.10

Gas Flow Distance (m)

Pe

rme

ab

ility

Ra

tio

(k/k

0)

t=1 s

t=100 s

t=1000 s

t=20000 s

Fig. 4. Example I: spatial and temporal distributions of permeability ratio

(k/k0).

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

2.2

0.0 1.0 2.0 3.0 4.0 5.0 6.0

Pore Pressure (MPa)

Perm

eabili

ty R

atio (

k/k

0)

K/Ks=1/3

K/Ks=1/10

K/Ks=0

Without Desorption

Palmer-Mansoori

Fig. 5. Example I: impacts of different bulk modulus ratio (K/Ks) and gas

desorption on permeability ratio (k/k0).

H. Zhang et al. / International Journal of Rock Mechanics & Mining Sciences 45 (2008) 1226–12361232

volumetric strain is more significant than the bulkmechanical volumetric strain. The contribution from thegrain volumetric strain is not obvious.

4.1.2. Gas storage

As shown in Eq. (37), the storage term consists of fourcontributions: from free gas, from gas absorption, fromcoal deformation, and from the combined effect ofsorption and deformation. Contributions from differentsources are presented in Fig. 3. As pore pressure depletes,94.6–97.1% of the storage coefficient is from gas absorp-

tion capability and 9.2–3.0% is from free gas storagecapacity. The contributions from the remaining twosources are below 4.5%, and can be neglected.

4.1.3. Permeability evolution

The spatial and temporal variations of the permeabilityratios are shown in Fig. 4. As the pore pressure declines,the permeability ratio increases with time. The permeabilityin the area near the right edge changes more rapidlythan the area far from this edge because the pressuregradient close to right edge is far greater. The steady stateis reached at about 20,000 s (�5 h) when the pressure isequal to 1 atm.

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0.0

1.0

2.0

3.0

4.0

5.0

6.0

0.00 0.02 0.04 0.06 0.08 0.10

Gas Flow Distance (m)

Pore

Pre

ssure

(M

Pa)

K/Ks=1/3, 1 s

K/Ks=1/3, 100 s

K/Ks=1/3, 1000 s

K/Ks=1/3, 20000 s

Without Desorption, 1 s

Without Desorption, 100 s

Without Desorption, 800 s

Fig. 6. Example I: spatial and temporal distributions of pore pressure

with and without gas desorption (K/Ks ¼ 1/3).

0.0

1.0

2.0

3.0

4.0

5.0

6.0

0.00 0.02 0.04 0.06 0.08 0.10

Gas Flow Distance (m)

Pore

Pre

ssure

(M

Pa)

K/Ks=1/3, 1s

K/Ks=1/3, 100 s

K/Ks=1/3, 1000 s

K/Ks=1/3, 20000 s

K/Ks=1/10, 1 s

K/Ks=1/10, 100 s

K/Ks=1/10, 1000 s

K/Ks=1/10, 12000 s

K/Ks=0, 1 s

K/Ks=0, 100 s

K/Ks=0, 1000 s

K/Ks=0, 10000 s

Fig. 7. Example I: impact of different bulk modulus ratio (K/Ks) on the

pore pressure distribution.

t=10 s

t=100 s

t=20000 s

t=0 s

Fig. 8. Example I: evolution of the simulated coal sample configuration

with gas desorption.

H. Zhang et al. / International Journal of Rock Mechanics & Mining Sciences 45 (2008) 1226–1236 1233

4.1.4. Impact of modulus ratios on permeability

As shown in Fig. 5, the permeability ratio (k/k0)increases due to gas desorption when pore pressuredecreases. When the ratio of bulk modulus (K/Ks) changesfrom 1/3 to 0, the highest permeability ratio varies from1.55 to 2.03. When the bulk modulus of coal grains (Ks) isassumed to be infinite, the simulation result is identicalwith the data calculated by the Palmer–Mansoori model. Ifthe gas desorption is neglected, the permeability ratiodrops linearly to 0.66.

4.1.5. Gas pressure distributions

The spatial and temporal variations of the gas pressureare shown in Figs. 6 and 7. These results demonstrate that

gas desorption has a much more significant impact on thegas pressure distribution than the bulk modulus ratios.

4.1.6. Coal deformation

As shown in Fig. 8, the coal deformation changes withtime due to the gas desorption. It shows that the coalsample shrinks with decreasing pore pressure in thehorizontal direction.The model results presented above reveal the character-

istics of the gas desorption from coal as evident in theexperiments [7,10]. One of these characteristics is that thegas sorption-induced volumetric strain plays an importantrole in the variation of coal porosity and permeability. Ourmodel results show that under the conditions of uniaxialstrain and constant overburden load, the influence from thecoal grain deformation can be neglected when estimatingvolumetric strain, porosity and pressure evolution. How-ever, the bulk modulus of grains may not be simply treatedas infinite when calculating permeability because the cubicrelationship between porosity and permeability. A slightchange in porosity may result in a much larger change inpermeability. If gas desorption is not considered, thepermeability may decrease linearly due to the increasingeffective stress. Otherwise, gas desorption may increase thepermeability and the net change in permeability iscontrolled by the opposite effects from effective stressand gas desorption. The pore pressure may be under-estimated dramatically if the effect of gas desorption isneglected. Although the permeability may decrease withoutgas desorption and cause gas to flow slowly, the storagecoefficient becomes much smaller and pore pressuredecreases very quickly. Therefore, the gas absorptioncapability dominates the storage coefficient.

4.2. Example II: gas desorption under constrained plane

strain

The new coupled model is applicable to variable loadingconditions. The following example is used to simulate the

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0.1

m0.1 m

p(0) = 6.2 MPa (Ω)

Fig. 9. Example II: simulation model of gas desorption from a coal

sample under the plane strain state with a constant overburden load.

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

2.2

0.0 2.0 4.0 6.0

Pore Pressure (MPa)

Perm

eabili

ty R

atio (

k/k

0)

K/Ks=1/3

K/Ks=1/10

K/Ks=0

Without Desorption

Palmer-Mansoori

1.0 3.0 5.0

Fig. 10. Example II: impacts of different bulk modulus ratio (K/Ks) and

gas desorption on permeability ratio (k/k0).

t=10 s

t=100 s

t=0 s

t=4000 s

Fig. 11. Example II: evolution of the coal sample configuration with gas

desorption.

H. Zhang et al. / International Journal of Rock Mechanics & Mining Sciences 45 (2008) 1226–12361234

coal sample under the constrained plane strain state asshown in Fig. 9.

The same geometrical model and the same materialproperties (Table 1) are applied as previous. The upper andright sides are unconstrained. The displacements at the leftand bottom sides are constrained in the horizontal andvertical directions, respectively. A distributed overburdenload of 6.9MPa is applied on the upper side and remainsunchanged during gas desorption. The initial gas porepressure in the coal is 6.2MPa and the pressure on the rightside remains at 101.325 kPa. Zero fluxes are specified on theother three boundaries.

The relations between gas pressure and permeabilityratio under different conditions are shown in Fig. 10. Asthe pore pressure decreases, the permeability decreasesfirst, then increases after the pressure comes to a criticalvalue of about 1MPa. The permeability ratio (k/k0) variesnot significantly when the ratio of bulk modulus (K/Ks)changes from 1/3 to 0. The biggest relative difference is6.7%. However, the maximum permeability ratios calcu-lated by using the Palmer–Mansoori model are significantlyhigher. The relative error is 141–153%. If the gasdesorption is neglected, the permeability ratio decreasesto 0.59 linearly. As shown in Fig. 11, the coal shrinks inboth directions when the gas pressure declines.

The impact of the different boundary conditions issignificant in this example although the property para-meters are the same as those in the Example I. Theoverburden loading and plane strain conditions increasethe effective stress and reduce the permeability in contrastto the uniaxially loaded condition in the first example. Thisdifference is the reason why the permeability calculated by

the Palmer–Mansoori model is much greater than ourmodel results.

4.3. Example III: gas desorption under the unconstrained

plane strain

In this example, we change the left boundary of ExampleI from the constrained horizontal displacement conditionto an unconstrained condition, as illustrated in Fig. 12. Allother conditions remain the same as Example II.The relations between gas pressure and permeability

ratio under different conditions are shown in Fig. 13. If gasdesorption is included, the permeability decreases initiallywhen pore pressure declines. Then it rebounds at thepressure of about 3MPa and increases to 0.91. Theinfluence of the bulk modulus ratio (K/Ks ¼ 1/3, 1/10, 0)is not significant. If gas desorption is not included, thepermeability ratio decreases linearly with decreasing pore

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0.1

m

0.1 m

p(0) = 6.2 MPa (Ω)

Fig. 12. Example III: simulation model of gas desorption from a coal

sample under the unconstrained plane strain state.

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

2.2

0.0 2.0 4.0 6.0

Pore Pressure (MPa)

Perm

eabili

ty R

atio (

k/k

0)

K/Ks=1/3

K/Ks=1/10

K/Ks=0

Without Desorption

Palmer-Mansoori

1.0 3.0 5.0

Fig. 13. Example III: impacts of different bulk modulus ratio (K/Ks) and

gas desorption on permeability ratio (k/k0).

t=5000 s

t=100 s

t=10 s

t=0 s

Fig. 14. Example III: evolution of the coal sample configuration with gas

desorption.

H. Zhang et al. / International Journal of Rock Mechanics & Mining Sciences 45 (2008) 1226–1236 1235

pressure. If the Palmer–Mansoori model is applied, thepermeability ratio increases throughout and the maximumdeviation from the current model is about 121%. As shownin Fig. 14, the coal shrinks both in the horizontal directionand in the vertical direction simultaneously when the gaspressure declines.

This example is analogous to Example II and alsodemonstrates that the loading and boundary conditionshave significant impact on model results. The permeabilityof our model is much less than that of the Palmer–Mansoori model. The difference indicates that the influenceof effective stress is underestimated due to the assumptionsin the Palmer–Mansoori model. Only one direction of

compressive strain is considered in the Palmer–Mansoorimodel, so that the impact of gas desorption is exaggeratedwhen neglecting the compressive strain in the otherdirection.

5. Conclusions

In this study, a new coupled gas flow and sorption-induced coal deformation finite element model is developedto quantify the net change in permeability, the gas flow,and the resultant deformation of the coal seam. Thecoupling between gas flow and coal deformation is realizedthrough a general porosity and permeability model. Thegeneral porosity model considers the principal controllingfactors, including the volume occupied by the free-phasegas, the volume occupied by the adsorbed phase gas, themechanical deformation-induced pore volume change, andthe sorption-induced coal pore volume change. Moreimportantly, these factors are quantified under in situstress conditions. A cubic relation between coal porosityand permeability is introduced to relate the coal storagecapability (changing porosity) to the coal transportproperty (changing permeability). The general porosityand permeability model is then implemented into the newcoupled gas flow and coal deformation finite elementmodel.The FE model has been applied to compare the

performance of the new model with that of the Palmer–Mansoori model. It is found that the Palmer–Mansoorimodel may produce significant errors if loading conditionsdeviate from the assumptions of the uniaxial straincondition and that of infinite bulk modulus of the coalgrain. The FE model has also been applied to conduct anumber of simulation examples. The model results haverevealed the characteristics of the gas desorption from coalas evident in the experiments reported in the literature.When the pore pressure declines during gas desorption,the net change in permeability accompanying the gas

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production is controlled competitively by the effects ofdeclining pore pressure and increasing shrinkage of coal.

Acknowledgment

This work is supported by the Australia ResearchCouncil under Grants DP0342446 and DP0209425 andby the Australian Department of Education, Science andTraining through the Australia-China Special Fund. Thissupport is gratefully appreciated.

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